Chi-Squared or Crosstabs
POLS 3316: Statistics for Political Scientists

Hypothesis Testing: Chi-square \(X^2\)

• Background
• Examples
• Practice

What is \(X^2\)

• Frequency distribution

• Used in hypothesis testing

• Categorical variables

• Ideal sample size 50- 1000

            + Less than 10 very unreliable

• Two hypothesis tests

            - Test of goodness of fit (1 variable)
            - **Test of independence (2 variables)*

\(X^2\) Test of independence

• Categorical variables

    + Testing if categories are related
    + Men/women, Republicans/Democracts, Northerners/Southerners, Black/White

• 2nd variable is the thing we look for a difference in

    + Pay, attitudes toward taxes, attitudes toward military service, victimization by police

\(X^2\) Test

• \(X^2\) tests the P(\(\frac{(Observed - Expected)^2}{Expected}\))
• The score is \(\frac{(Observed - Expected)^2}{Expected}\))
• Expected = \(\frac{row total x column total}{sample size}\)
• H0 (null hypothesis): in the population, the two categorical variables are independent not related
• H1 (alternative/research hypothesis): in the population, the two categorical variables are dependent related
• p-value is P(random effect > computed effect) so smaller is better

Chi Square Visualization

Visualization of Chi-Square

Not a normal distribution

Chi Square Distribution

• the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables

Example